Is this a fair game? Explain why or why not.

Question

You toss a circular coin randomly onto the Adversary’s† infinite checkerboard. That is, it is as likely to turn up in one location as any other location. The coin’s diameter is exactly half the length of the sides of the squares of the checkerboard, and the coin does not come to rest on its edge. If the coin touches a side of one or more of the squares, the Adversary keeps it; if it does not, the Adversary returns the coin and gives you three more coins. 

Answer 1

Best way to calculate probability in this case is by area.  Let L be the length of one side of each square.  We see that with a L/2 diameter coin, there is a square of side length L/2 centered in each square that the center of the coin would have to fall within in order to not touch the sides of the checkers square.  The area of the whole square is L² and the area of the smaller square is (L/2)² = L²/4.  Therefore, the probability of not touching an edge is 1/4, while the probability of touching an edge is 3/4.  

 

The expected outcome can be calculated by multiplying each probability by the amount you gain with that outcome, so:

 

1/4*3 + 3/4*(-1) = 3/4 - 3/4 = 0

 

Therefore, the game is fair.